Gödel's Proof

Ernest Nagel + James R. Newman

An early attempt to explain Gödel's Incompleteness Theorem to a broader
audience, this little 1958 book by Nagel and Newman is still one of the
nicest presentations. Anyone who has studied mathematical logic will
probably be able to read it in a sitting, but it should be accessible,
with some work, to those with only school mathematics. Familiarity with
a formal deductive system such as Euclidean geometry would definitely
help, though that's something that can't be taken for granted even with
mathematics students these days.

For the first sixty pages Nagel and Newman combine history — Euclid,
Hilbert and attempts to formalise mathematics, Russell and Whitehead's
Principia Mathematica — with an explanation of the necessary
preliminaries — the problem of consistency, absolute proofs of
consistency, the codification of formal logic, the difference between
mathematics and meta-mathematics (reasoning about mathematics), and
the key idea that "meta-mathematical statements about a formalized
arithmetical calculus can ... be represented by arithmetical formulas
within the calculus".

In thirty pages they then present the proof itself: Gödel numbering,
arithmetizing meta-mathematics, and the actual construction. This is a
drastic simplification of the full proof, obviously, but it is enough
to, as they put it, "afford the reader glimpses of the ascent and of
the crowning structure". They close with some brief comments on the
broader philosophical implications of Gödel's Theorem.